Answer
$f^{\prime}(x)=1+0.5x^{-0.5}$
Work Step by Step
SUMMARY:
The Power Rule$:\ \ \ [x^{n}]^{\prime}=nx^{n-1 } $
Sum Rule: $\ \ \ \ \ \ [f\pm g]^{\prime}(x)=f^{\prime}(x)\pm g^{\prime}(x) $
Constant Multiple Rule:$\ \ \ [cf]^{\prime}(x)=cf^{\prime}(x) $
Constant times x:$\ \ \ \displaystyle \frac{d}{dx}(cx)=c $
Constant:$\displaystyle \ \ \ \ \ \frac{d}{dx}(c)=0 $
--------------------------------
$f^{\prime}(x)=[x+x^{0.5}]^{\prime}=... $Sum Rule,
$=[x]^{\prime}+[x^{0.5}]^{\prime}=$... individually:
$[x]^{\prime}=[1\cdot x]^{\prime} ...$Constant times x$...=1$
$[x^{0.5}]^{\prime}$=...power rule...$=0.5x^{-0.5}$
So
$f^{\prime}(x)=1+0.5x^{-0.5}$